Instituto de Física USP PASI 2012 UBA
Theories with Compact Extra dimensions Part II Generating Large Hierarchies with ED Theories Warped Extra Dimensions
Warped Extra Dimensions One compact extra dimension. Non-trivial metric induces small energy scale from Planck scale! (L. Randall, R. Sundrum). AdS 5 k Planck TeV ke k L L Geometry of extra dimension generates hierarchy exponentially! Λ TeV M Planck e k L with k the curvature
Warped Extra Dimensions Warped 5D metric in RS ds 2 = e 2κ y η µν dx µ dx ν dy 2 Compactified on S 1 /Z 2 with L = πr AdS 5 κ κ e κ π R and k < M P, AdS 5 curvature. For kr (11 12) κ e κπr O(TeV). 0 y πr
Warped Extra Dimensions Solving the Hierarchy Problem: If Higgs localized at y = πr S H = d 4 x πr 0 dy g δ(y πr) [ g µν µ H ν H λ ( H 2 v0 2 ) ] 2 Warp factors e ky appear in g µν and g. [ S H = d 4 x e 2kπR η µν µ H ν H e 4kπR λ ( H 2 v0 2 ) ] 2 Canonically normalize H e kπr H H
Warped Extra Dimensions What we get now is [ ( ) ] 2 S H = d 4 x η µν µ H ν H λ H 2 e 2kπR v0 2 If v 0 M P, then new scale in exponentially suppressed v = e kπr v 0 Choosing kr O(10) gets us v weak scale.
Solving the Hierarchy Problem in AdS 5 Metric in extra dimension small energy scale from M P (Randall-Sundrum) ds 2 = e 2κ y η µν dx µ dx ν dy 2 AdS 5 Corrections to m h OK If Higgs close to TeV brane M P Planck TeV M P _ kl e Need Higgs IR localization L
Warped Extra Dimensions If only gravity propagates in the bulk (SM fields on TeV brane): = Kaluza-Klein graviton tower Zero-mode graviton G (0) localized toward the Planck brane. This is why gravity is weak! G (0) couples to SM fields as 1/M 2 P First few KK graviton excitations localized toward TeV brane They couple strongly (as (1/TeV) 2 to fields there. E.g.: Drell-Yan at hadron colliders q G (n) e+ q e
The Origin of Flavor In the SM, flavor comes from dim-4 ops. For instance for quarks L Y = Y U ij Q i Hu j + Y D ij Q i Hd j, and similarly for leptons. Masses arise after EWSB: H = v/ 2 m ij D = Y D ij v 2 But what is the origin of the Yukawa couplings Y U, Y D, etc.?
The Origin of Flavor Building a theory of flavor in extensions of the SM is not easy. Scale of EWSB is TeV scale separation of scales with flavor. Supersymetry: Flavor is generated at very high scale, possibly GUT scale. (Avoiding flavor vioation from running effects is hard!) Technicolor: TC gets strong at TeV EWSB. Flavor comes from ETC gauge bosons, with M (100 1000) TeV. In all cases, is very hard to accommodate the spectrum of fermion masses naturally.
Bulk Life in WED In original proposal, only gravity propagates in 5D bulk. Allowing gauge fields and matter to propagate in the bulk opens many possibilities: models of EWSB, flavor, GUTs, etc. Bulk Randall-Sundrum models: Choose the gauge symmetry in the bulk: The electroweak SM gauge group has to be enlarged to avoid large corrections to EWP observables: SU(2) L SU(2) R U(1) X Write theory in the bulk and expand in Kaluza-Klein modes to get the effective 4D description.
Gauge Fields in the 5D bulk A gauge field A M (x µ, y) propagating in the extra dimensional bulk: S A = 1 d 4 xdy g F MN F MN, 4 with g = det[g MN ] = e 4σ, σ k y. Field strength defined as F MN M A N N A M + ig 5 [A M, A N ] The KK decomposition is A µ (x, y) = 1 A (n) µ (x)χ (n) (y) 2πR n=0
Gauge Fields in the 5D bulk χ (n) (y): wave-function of the n-th KK mode in the extra dimension. Imposing the EoM (Eüler-Lagrange) on A M (x µ, y) we get ( y e 2σ y χ (n)) = mnχ 2 (n) They satisfy the ortho-normality condition 1 πr dy χ (m) χ (n) = δ mn 2πR πr Solving this differential equation gives χ (n) (y).
Gauge Fields in the 5D bulk The solutions are of the form χ (n) (y) = eσ [ J 1 ( m n N n κ eσ ) + α n Y 1 ( m ] n κ eσ ) where N n is a normalization factor, and again σ = k y. In the interval [0, πr], χ (n) peaks πr for low values of n. first few KK modes of gauge bosons are localized near the TeV brane.
Gauge Fields in the bulk - KK Masses The constants α n, and the KK masses are determined by the boundary conditions on the Planck and TeV branes. The masses of the first few KK modes are m n (n O(1)) πκe κπr I.e. for appropriate choice of κr 1st KK excitations are O(TeV). To be expected: what is localized near the TeV brane has TeV-like masses!
Fermions in Warped Extra Dimensions The action for fermion fields Ψ(x µ, y) in the bulk S f = d 4 x dy { i g 2 Ψˆγ [ M D M D ] } M Ψ sgn(y)m f ΨΨ with the covariant derivative given by D M M + 1 8 [γα, γ β ] V N α V βn;m and ˆγ M V M α γ α, with V M α = diag(e σ, e σ, e σ, e σ, 1) the inverse vierbein. To be natural, we need M f O(1)k, with k M Planck, the only scale.
Fermions in Warped Extra Dimensions Although Ψ(x µ, y) is not a chiral fermion, we can still write Then the KK expansion is Ψ L,R 1 2 (1 γ 5) Ψ L,R (x, y) = 1 2πR n=0 ψ L,R n (x)e 2σ fn L,R (y) with fn L,R (y) the wave-function of the n-th KK fermion in the extra dimension.
Fermions in Warped Extra Dimensions Demanding that ψn L,R (x) be usual 4D fermions, we obtain ( y M f ) fn L (y) = M n e σ fn R (y) ( y + M f ) fn R (y) = M n e σ fn L (y) Zero mode fermions: M 0 = 0. Solving the differential eqns. we get: f R,L kπr (1 ± 2c L,R ) 0 (y) = e kπr(1±2cl,r) 1 e±c L,R k y where we defined M f = c f k. the bulk mass parameter c f O(1), defines the localization of the fermion in the extra dimension.
Fermions in Warped Extra Dimensions To see how the localization depends on c we need to normalize the fermions canonically. Then the zero-mode fermion wave-function is F L ZM(y) = 1 2πR f L 0 (0)e ( 1 2 c L) ky If c L > 1/2 fermion localized near y = 0, Planck brane. If c L < 1/2 fermion localized near y = πr, TeV brane.
Flavor Models in WED O(1) flavor breaking in bulk can generate fermion mass hierarchy: C>1/2 C<1/2 Higgs C=1/2 0 Fermions localized toward the TeV brane can have larger Yukawas, Those localized toward the Planck brane have highly suppressed ones. πr
Yukawa Couplings For instance, assuming a TeV-brane localized Higgs, the Yukawa couplings are S Y = d 4 x dy g λ5d ij Ψ i (x, y)δ(y πr)h(x)ψ j (x, y) 2 M 5 Then, the 4D Yukawa couplings are ( ) λ 5D ij k (1/2 c L ) (1/2 c R ) Y ij = e kπr(1 2cL) 1 e kπr(1 2cR) 1 ekπr(1 c L c R ) M 5
Yukawa Couplings The Yukawa couplings as a function of c L, assuming c R = 0.4 (i.e. TeV localized right-handed fermion):
The Bulk RS Picture SU(2) SU(2) L x R x U(1) Light Fermions t b L t R Higgs or Higgless Models of EWSB and Flavor 0 L EWPC: T OK, but S N/π at tree-level M KK > (2 3) TeV Z bb require discrete symmetry (L R) (Agashe, Contino, Da Rold, Pomarol) Potentially important bounds and/or effects from flavor violation
Dynamical Origin of the Higgs Sector What localizes the Higgs to/near the IR/TeV brane? Gauge-Higgs Unification Zero-mode Fermion Condensation Higgsless
Gauge-Higgs Unification in AdS 5 If there is a Higgs: what is its dynamical origin? Or why is it localized towards the TeV brane? Gauge field in 5D has scalar A 5 To extract H from A 5 need to enlarge SM gauge symmetry. E.g. SU(3) SU(2) U(1) by boundary conditions: A µ : A 5 : (+, +) (+, +) (, ) (+, +) (+, +) (, ) (, ) (, ) (+, +) (, ) (, ) (+, +) (, ) (, ) (+, +) (+, +) (+, +) (, ) Higgs doublet from A 5 = A a 5 ta
Gauge-Higgs Unification in AdS 5 To build realistic models of EWSB from AdS 5 : Isospin symmetry: need SO(4) U(1) X in bulk SO(5) U(1) X SO(4) U(1) X by BCs (Agashe, Contino, Pomarol) Higgs is 4 of SO(4): 4 d.o.f. complex SU(2) L doublet Gauge bosons and fermions in complete SO(5) multiplets Implementing additional symmetry to protect Z b b spectrum of KK fermions, lighter than KK gauge bosons. (Contino, Da Rold, Pomarol)
Gauge-Higgs Unification in AdS 5 E.g.: Fermions can be 5 2/3 = (2, 2) 2/3 (1, 1) 2/3 or 10 2/3 = (2, 2) 2/3 (1, 3) 2/3 (3, 1) 2/3 to satisfy custodial + L R symmetry. BCs masses of KK fermions tend to be light (because top is heavy)
Gauge-Higgs Unification in AdS 5 Signals: Rich gauge boson spectrum, at few TeV Light KK fermion spectrum: could be as light as 500 GeV Very distinctive signals: E.g. b-type KK fermion tw 4W s + 2b signals (Dennis, Servant, Unel, Tseng) Enhanced t 1 pair production through KK gluon (Carena, Medina, Panes, Shah, Wagner)
Flavor Violation in AdS 5 Models KK Gauge Bosons couple stronger to heavier fermions KK Gauge Bosons light fermions heavy fermions 0 L Tree-level flavor violation is hierarchical: Only important with the heavier generations
Flavor Violation: The Good If ZM fermion is localized close enough to IR can interaction with KK gauge boson (e.g. KK gluon) be strong enough to condense fermions? Need F F F is ZM quark Scale of condensation is M KK 1 TeV m F (500 600) GeV 4th Generation Condensation in AdS 5
EWSB from Fourth-Generation in AdS 5 Need 4th-generation strongly coupled to new interaction If 4th-generation propagates in AdS 5 bulk and is highly localized on the TeV brane (G.B., Da Rold) 4th-generation quarks are strongly coupled to KK gluon:
EWSB from Fourth-Generation in AdS 5 Solution to the gap equation: If g U > g crit. U, ŪLU R 0 This implies Electroweak Symmetry Breaking Dynamical m U We can also write an effective theory at low energy for the Higgs.
Predictions for m U, m H For M KK = 3 TeV, we get m U 600 GeV m h 900 GeV Higgs naturally TeV-brane localized Fermion masses (other than m U ): From bulk Higher Dimensional Ops.
Two Higgs Doublets from 4G Condensation If both U and D 4th gen. quarks condense Two-Higgs Doublet model (GB Haluch 11) PQ Symmetry pseudo-scalar A is light: m A (25 130) GeV. Scalars are heavy (h, H, H ± ) in the (550 700) GeV range Phenomenology at LHC interesting (GB, Haluch, Matheus 11) But current LHC bounds on 4G quarks closing in m Q4 > 500 GeV
Flavor Violation: The Bad Constraints from low energy/flavor physics Potentially, deviations in CP asymmetries in B decays b sl + l m Bs D 0 D 0 mixing But for M KK > 2 TeV, and/or adjusting the down-quark rotation matrices D L,R, these bounds can be evaded.
Single Top Production at High Invariant Mass (Aquino, GB, Eboli 07) q _ q g q G 00 11 00 11 00 11 t _ c g t Utc Signal: t + jet, very large invariant mass (> 1.5 TeV). Use t blν. Typically few hundred fb 1.
Flavor Violation: The Ugly But one flavor observable is hard to get around: ɛ K Mixed Chirality operators d R s L d L s R have enhancement of ( mk m s ) 2 η 5 1 > 100 Bound from ɛ K is M KK > 30 TeV!! Need flavor symmetries to protect from this.
Future Avenues Build AdS 5 models with flavor symmetries protecting from ɛ K bounds Abandon AdS 5 dogma: Keep the good features Abandon 5D metric: Deconstruct AdS 5. Resulting 4D Theories have much less flavor violation far from the continuum (GB, Fonseca, Lima 12)